The application of the quantum Fourier transform (QFT) within the field of quantum computation has been manifold. Shor’s algorithm, phase estimation and computing discrete logarithms are but a few classic examples of its use. These initial blueprints for quantum algorithms have sparked a cascade of tantalizing solutions to problems considered to be intractable on a classical computer. Therefore, two main threads of research have unfolded. First, novel applications and algorithms involving the QFT are continually being developed. Second, improvements in the algorithmic complexity of the QFT are also a sought after commodity. In this work, we review the structure of the QFT and its implementation. In order to put these concepts in their proper perspective, we provide a brief overview of quantum computation. Finally, we provide a permutation structure for putting the QFT within the context of universal computation.
- quantum Fourier transform
- quantum computation
- quantum circuit
- unitary operators
- permutation operators
The quantum Fourier transform (QFT) has been applied in a number of different contexts within the field of quantum computation [1, 2, 3]. As this operator is central to so many quantum algorithms, a major thrust of current research is directed toward its efficient implementation [4, 5, 6, 7, 8, 9]. The QFT calculation is, to a degree, based upon the discrete Fourier transform (DFT) where, given a discrete sequence
of length , the DFT of can be computed as
with DFT matrix elements
Since the DFT matrix is , the computational complexity of computing is . If the input sequence length of the input sequence can be written as (i.e. a power of two for some positive integer, ), there exist fast Fourier transform (FFT) implementations that can compute with complexity. While there are other FFT implementations that do not require , the ‘radix-2’ implementation will be the starting point as it is relevant when introducing quantum computational bases. Before elevating the DFT to its quantum description, in Section 2 we will take a brief tour of quantum computation in order to provide some necessary context. We will then, in Section 3, develop the QFT operator and discuss its quantum implementation. Finally, in Section 4, we will discuss the QFT in the context of universal computation and its formulation in terms of permutation matrices.
2. Quantum computation
A starting point for quantum computation begins with choosing a qubit representation for the computational basis 
This qubit basis forms a complete orthonormal set so that any single qubit quantum mechanical state can be written as the linear superposition
where the coefficients and are complex scalars. If represents the Hermitian conjugate of , according to quantum mechanics, the inner product
is normalized so that represents a probability density function. This implies that, at any given instance in its time evolution, a quantum system can simultaneously be in the logical states and with their associated probabilities and . This is in stark contrast to classical digital computation whose operations must always exclusively evaluate to a value of either 0 or 1. Quantum computation allows an algorithm to simultaneously visit
2.1 Unitary operators
The time evolution operator associated with a quantum system must be unitary meaning that
where is the conjugate transpose of . A major implication of this requirement is that the forward time system evolution must (at least mathematically) be reversible. This requirement, in turn, constrains computations that are implemented by quantum operators to be
The reader can check that these are all unitary. As a simple example of how to apply such operators, consider the action of on the basis vector
where and are ‘swapped’, indicating a form of logical inversion. is a Hadamard transform (i.e. a DFT for a sequence of length N=2). , and are Pauli matrices. is a generalization of and . While these are single quhit operators, the next sections discuss how they can be extended to the multiple qubit case. Amazingly, this set of quantum operators can be applied to devise some very powerful quantum algorithms (e.g. QFT computation) [3, 10].
2.2 Tensor product (Kronecker product)
The Kronecker product of an matrix with a matrix is defined to be
Furthermore, assuming the dimensions are compatible for matrix multiplication, the following identity often proves useful
for matrices .
The computational basis can be extended to any number of qubits using the tensor product. For example, if two qubits are required for the computational space, using Eq. (2), the basis becomes
To generalize this example for qubits, the set of computational basis vectors can, for the sake of brevity, be labeled in base 10 as
On the other hand, in order to highlight the qubit values, this basis can equivalently be expressed in base 2 as
where for . In other words, represents the binary expansion
for the basis vector . We have chosen this bit index ordering as it will prove convenient for the QFT formulation in the next section. An equally acceptable (and, quite typical) bit index convention for an qubit system could, for example, be .
Eq. (15) tells us that the qubit basis is derived from the tensor product of single qubits. This is important to keep in mind in order to avoid confusion when using the symbol . For example, when using qubit, in decimal is equivalent to in binary; however,, when using qubits, in decimal is equivalent to in binary. Hence, the number of qubits is the anchor for the relationship between Eq. (14) and Eq. (15). Assuming qubits, there are basis vectors that can be used to construct a quantum state. Hence, all basis vectors will simultaneously evolve with their associated probabilities; again, this is the source of quantum parallelism.
2.3 Quantum circuits
One particularly useful application of Eq. (12) arises when building up qubit quantum circuits (i.e. schematic depictions of quantum operations on qubits). For instance, assume a two qubit system where two unitary operators and act on single qubits as
and the result is desired to be combined as
Eq. (12) tells us that this action is equivalent to
However, by construction, . Therefore,
making it straightforward to develop multiple qubit quantum systems from unitary operators. The schematic representation of is show in Figure 1.
With the groundwork laid for multiple qubits, it becomes possible to introduce more unitary operators that facilitate reversible computation. For example, the controlled NOT (CNOT) function can be phrased as a two qubit reversible XOR operator
where represents the control bit, represents the target XOR function and . This operator is a permutation matrix that is consistent with Table 1 in that it swaps the and qubits. The XOR operation, by itself, can act as an irreversible controlled NOT operation. For the sake of quantum computation, the CNOT operator is unitary and a reversible XOR function is achieved because the control bit is preserved from input to output.
There exist powerful tools for the simulation of quantum operations (referred to as ‘
For the sake of this work, we point out that an equally valid interpretation of the quantum CNOT function can be realized if the roles of the control and target are interchanged where (see Table 2). In this case the CNOT operator becomes
which is a permutation matrix that swaps the and qubits and corresponds to the circuit in Figure 3.
We shall have more to say about this implementation in the following sections. For now, with this brief overview of quantum computation, we can now introduce the quantum Fourier transform.
3. The quantum Fourier transform
It should be clear that the DFT matrix in Eq. (2) is unitary where
and is the Hermitian conjugate of . Because of this unitarity, the potential for using the DFT within the context of quantum computation naturally follows. However, such an application requires a decomposition involving tensor products of unitary operations typically applied in quantum computation. As with the FFT, the choice of the decomposition dictates the algorithmic complexity. There is much introductory literature available regarding the QFT [3, 12, 13, 14]. Given a specific quantum algorithm where the QFT is applied, current research endeavors reside in attempts to improve the computational complexity [4, 7, 9, 15, 16].
The QFT matrix is defined as
For example, with and n = 1, we recover the Hadamard matrix
or, for n = 2,
As expected, this operator is unitary where, with
it should be clear that
In general, given a state vector
the QFT operates on to form
Given this result, let us consider the QFT of a single qubit basis vector where . First, observe that while
This leads to the result that
3.1 QFT qubit representation
To forge a path toward efficient implementation, it is important to recognize how Eq. (33) can be decomposed into a set of operators relevant to quantum computation (see Section 2.1). First, consider the single qubit case,
Then, for each qubit state , it follows that
as expected since for the single qubit case. Hence, it should be nn surprise that the contribution to Eq. (10) should be a Hadamard gate.
To handle the phase factors in the other contributions to the tensor product (where ), the keen eye will recognize that the terms could lead to a unitary quantum mechanical operator. Before leveraging this observation in a QFT algorithm, it will be helpful to consider the qubit representation . As the index ranges from to , the index in the term experiences successive divisions by (i.e. successive right shifts of its binary representation by one bit):
Since these values appear in the phase factor, the integer parts will only result in integer multiples of and can therefore be discarded. Eq. (33) can then be expressed as
It is often this version of the QFT that is used as a starting point for quantum circuit implementation when .
As an example, consider the two qubit case where and , then
If we let , then
which corresponds to the column entries in Eq. (26). If not already obvious, it should be emphasized that the tensor product is
3.2 Quantum implementation
Based upon Eq. (37), it is sensible to introduce an iterable version of the operator introduced in Section 2.1:
Furthermore, because each qubit contribution contains phase terms involving the binary expansion of , one approach to addressing these interactions is to introduce a controlled version of :
This operator can be used to induce the correct phase factor as follows. Assume is the target/control structure for single qubits were in the binary representation of . Then, the following holds true
Hence, the control bit determines when to introduce the phase factor involving the target bit.
The goal of this section is to introduce enough nomenclature in order to put the next section of this work in context. The reader is encouraged to visit the provided references in order to fill in the details of a generalized quantum circuit that can implement an qubit QFT. For now, we provide an qubit example to illustrate an algorithm for performing the QFT. Whatever principled series of operations is chosen, the goal of the quantum algorithm (and, hence, the associated quantum circuit) is to reproduce Eq. (11). Starting with ,
Apply to so thatE43
Apply to target qubit controlled by . This yields
Comparing this result with either Eq. (33) or Eq. (37), it is clear that this algorithm, derived using quantum reversible operators, recovers the QFT from Eq. (38) with one slight difference: the bit ordering is reversed. Given qubits, it is possible to apply swaps using, for example, tensor products involving an operator (see Section 2.1) in order to reverse the bit order. Such bit reversal permutations are reminiscent of the radix-2 FFT algorithm. If one generalizes this algorithm to qubits, it can be shown that the algorithmic complexity is . With , this is a considerable improvement over for the radix-2 FFT. However, algorithmic improvements and variations have been developed that can further reduce QFT complexity to [9, 15].
4. QFT permutations
Universal computation, by its very nature, must involve some set of permutation operators [17, 18, 19, 20]. As with other universal gates applied in quantum computation, in this section, we show that the QFT can generate operators that have the properties of a permutation. Consider a successive application of the QFT such as and let us analyze the matrix elements of such an operation:
For an qubit system , it should be clear that is a permutation operator that leaves the position of unchanged and inverts the order of the remaining qubits to form . For example, the CNOT operator in Eq. (22) is equal to for
having properties similar to that of a Sylvester shift matrix (i.e. a generalization of a Pauli matrix). It is sensible that a CNOT operation followed by a CNOT operation should result in the identity operation and, hence, that (i.e. a double inversion recovers the original qubit sequence). These results can be generalized for any . For example, with , Eq. (46) becomes
which, after the appropriate sequence of swaps, can be transformed into a Toffoli (CCNOT) gate. Hence, can be thought of as a generalization of swap permutation operators and the QFT can be phrased as its square root. For example, it is common to define a two qubit swap operator as
along with its square root
In a similar manner, Eq. (46) leads us to the following
it follows that
where is a QFT matrix.
In addition, given that we have the following
These results indicate a deeper connection between universal computation, permutations and the QFT. Furthermore, decomposing the QFT calculation into a product of permutations indicates a potential for reducing the computational complexity of QFT implementations.
In this work, we have revisited the quantum Fourier transform which is central to many algorithms applied in the field of quantum computation. As a natural extension of the discrete Fourier transform, the QFT can be implemented using efficient tensor products of quantum operators. Part of the thrust of current research deals with reducing the QFT computational complexity. With this goal in mind, we have phrased the QFT as a permutation operator. Future research will be directed toward quantum circuit implementation using QFT permutation operators within the context of universal computation.
This research is funded by a grant from the National Science Foundation NSF #1560214.